Trotter Kato Theorem for Bi Continuous Semigroups and Approximation of PDEs

TL;DR

This paper extends the Trotter Kato theorem to bi continuous semigroups, offering a framework for analyzing PDE approximations under weaker topologies, demonstrated through heat equation examples.

Contribution

It introduces a novel formulation of the Trotter Kato theorem for bi continuous semigroups applicable to PDE approximation with weaker topologies.

Findings

Framework for convergence analysis under locally convex topologies

Application to heat equation with infinite boundaries

Enhanced understanding of PDE numerical approximations

Abstract

In this paper, we introduce formulations of the Trotter Kato theorem for approximation of bi continuous semigroups that provide a useful framework whenever convergence of numerical approximations to solutions of PDEs are studied with respect to an additional locally convex topology coarser than the norm topology to treat the lack of the strong continuity. Applicability of our results is demonstrated using a heat equation with infinite boundaries.